## Master Program "Applied Mathematics and Stochastics"

## Numerical Statistical Modelling

This is a 2-year English-taught programme at the Master level. It is aimed at training and developing research skills in Numerical Modelling, Monte Carlo Methods and in their applications.

The Programme is research-oriented. Each semester a student is supposed to devote a half of time to a research project, to be done at Institute of Computational Mathematics and Mathematical Geophysics. Results of this research project must be presented in a form of a Master Dissertation at the end of 2nd year of studies.

Topics of lectures and research includes, but are not limited to

– numerical modelling of random values and processes,

– solution of ordinary and stochastic differential equations,

– theory of radiation transfer,

– stochastic simulation methods in applied mathematics, physics, meteorology, economics (including financial mathematics),

– discrete-stochastic numerical methods,

– modern computer technologies in stochastic simulation, etc.

Lecture courses and research supervision are provided by professors of the Novosibirsk State University and leading researchers from institutes of the Russian Academy of Sciences. They all are members of the worldwide known scientific school on numerical stochastic simulation headed by Professor G.A.Mikhailov.

## Inverse and Ill-Posed Problems: Theory, Numerics and Applications

The main purpose and characteristic feature of this program is the accessibility of presentation and an attempt to cover the rapidly developing areas of the theory, numerical methods and applications of inverse and ill-posed problems as completely as possible.

In direct problems of mathematical physics, researchers try to find exact or approximate functions that describe various physical phenomena such as the propagation of sound, heat, seismic waves, electromagnetic waves, etc. In these problems, the media properties (expressed by the equation coefficients) and the initial state of the process under study (in the nonstationary case) or its properties on the boundary (in the case of a bounded domain and/or in the stationary case) are assumed to be known. However, it is precisely the media properties that are often unknown. This leads to inverse problems, in which it is required to determine the equation coefficients from the information about the solution of the direct problem. Most of these problems are ill-posed (unstable with respect to measurement errors). At the same time, the unknown equation coefficients usually represent important media properties such as density, electrical conductivity, heat conductivity, etc. Solving inverse problems can also help to determine the location, shape, and structure of intrusions, defects, sources (of heat, waves, potential difference, pollution), and so on. Given such a wide variety of applications, it is no surprise that the theory and numerical methods of inverse and ill-posed problems has become one of the most rapidly developing areas of modern science. Today it is almost impossible to estimate the total number of scientific publications that directly or indirectly deal with inverse and ill-posed problems. However, since the theory, numerical methods are relatively young, there are many terms are still not well-established, many important results are still being discussed and attempts are being made to improve them. New approaches, concepts, theorems, methods, algorithms and practical problems are constantly emerging.

Main topics:

- Ill-Posed problem concept

- Inverse problems classification

- Regularization methods

- Applications: Geophysics, Medicine, Biology, Finance, Social sciences, Big Data, Data Mining, Machine Learning, Image Processing.

## Discrete Mathematics and Combinatorial Optimization

During the last century the discrete and computational mathematics got
boosting development which is connected mainly with the widespread involvement of
computers and computer technologies into the everyday life. This required the
development of new methods of studying, processing, transmitting, storing,
protecting and analyzing the large volumes of information and also training
the specialists knowing all these methods. A modern specialist in discrete
mathematics and combinatorial optimization has to possess a deep knowledge
of graph theory, scheduling theory, coding theory, cryptography, operations
research, data analysis methods, mathematical modeling, linear, convex and
integer programming, complexity theory, algorithms development and analysis,
etc.

The purpose of the Master Educational Programme "Discrete Mathematics and Combinatorial
Optimization" is training highly qualified specialists in the following areas:

– Operations research

– Optimization methods

– Discrete analysis

– Graph theory

– Scheduling theory

– Coding theory

– Algorithms

– Data analysis.

Such specialists should be capable of working in research institutes, universities, programming firms, engineering companies, etc.

The training is based on requirements of the State Educational Standard 01.04.02.